Properties

Label 9800.2.a.cj.1.4
Level $9800$
Weight $2$
Character 9800.1
Self dual yes
Analytic conductor $78.253$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9800,2,Mod(1,9800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9800.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(78.2533939809\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.43449.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 7x^{2} + 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1400)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.04673\) of defining polynomial
Character \(\chi\) \(=\) 9800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.04673 q^{3} +1.18910 q^{9} +O(q^{10})\) \(q+2.04673 q^{3} +1.18910 q^{9} -3.56405 q^{11} +2.90761 q^{13} +3.71851 q^{17} -3.57939 q^{19} -3.10880 q^{23} -3.70642 q^{27} +1.57939 q^{29} -0.764171 q^{31} -7.29465 q^{33} -7.00107 q^{37} +5.95109 q^{39} +7.07812 q^{41} -10.9390 q^{43} -7.37602 q^{47} +7.61078 q^{51} +3.32822 q^{53} -7.32604 q^{57} +8.14019 q^{59} -5.93793 q^{61} -14.0021 q^{67} -6.36286 q^{69} -14.5305 q^{71} +13.2216 q^{73} +2.98466 q^{79} -11.1533 q^{81} -10.8923 q^{83} +3.23258 q^{87} +4.71958 q^{89} -1.56405 q^{93} +10.6575 q^{97} -4.23801 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{3} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{3} + 5 q^{9} + 4 q^{13} + 7 q^{17} - 10 q^{19} - 12 q^{27} + 2 q^{29} - 14 q^{31} - 2 q^{33} + 2 q^{37} - 10 q^{39} - 4 q^{41} - 15 q^{43} + 15 q^{47} + 5 q^{51} + 10 q^{53} + 19 q^{57} - q^{59} - 25 q^{61} + 4 q^{67} - 16 q^{69} - 20 q^{71} + 2 q^{73} + 2 q^{79} + 24 q^{81} - 26 q^{83} - 13 q^{87} - 19 q^{89} + 8 q^{93} + 6 q^{97} - 51 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.04673 1.18168 0.590840 0.806789i \(-0.298797\pi\)
0.590840 + 0.806789i \(0.298797\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.18910 0.396367
\(10\) 0 0
\(11\) −3.56405 −1.07460 −0.537301 0.843391i \(-0.680556\pi\)
−0.537301 + 0.843391i \(0.680556\pi\)
\(12\) 0 0
\(13\) 2.90761 0.806426 0.403213 0.915106i \(-0.367893\pi\)
0.403213 + 0.915106i \(0.367893\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.71851 0.901871 0.450935 0.892557i \(-0.351090\pi\)
0.450935 + 0.892557i \(0.351090\pi\)
\(18\) 0 0
\(19\) −3.57939 −0.821168 −0.410584 0.911823i \(-0.634675\pi\)
−0.410584 + 0.911823i \(0.634675\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.10880 −0.648229 −0.324114 0.946018i \(-0.605066\pi\)
−0.324114 + 0.946018i \(0.605066\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −3.70642 −0.713301
\(28\) 0 0
\(29\) 1.57939 0.293285 0.146642 0.989190i \(-0.453153\pi\)
0.146642 + 0.989190i \(0.453153\pi\)
\(30\) 0 0
\(31\) −0.764171 −0.137249 −0.0686245 0.997643i \(-0.521861\pi\)
−0.0686245 + 0.997643i \(0.521861\pi\)
\(32\) 0 0
\(33\) −7.29465 −1.26983
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.00107 −1.15097 −0.575484 0.817813i \(-0.695186\pi\)
−0.575484 + 0.817813i \(0.695186\pi\)
\(38\) 0 0
\(39\) 5.95109 0.952937
\(40\) 0 0
\(41\) 7.07812 1.10542 0.552708 0.833375i \(-0.313594\pi\)
0.552708 + 0.833375i \(0.313594\pi\)
\(42\) 0 0
\(43\) −10.9390 −1.66818 −0.834091 0.551627i \(-0.814007\pi\)
−0.834091 + 0.551627i \(0.814007\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.37602 −1.07590 −0.537951 0.842976i \(-0.680802\pi\)
−0.537951 + 0.842976i \(0.680802\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 7.61078 1.06572
\(52\) 0 0
\(53\) 3.32822 0.457166 0.228583 0.973524i \(-0.426591\pi\)
0.228583 + 0.973524i \(0.426591\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −7.32604 −0.970357
\(58\) 0 0
\(59\) 8.14019 1.05976 0.529881 0.848072i \(-0.322237\pi\)
0.529881 + 0.848072i \(0.322237\pi\)
\(60\) 0 0
\(61\) −5.93793 −0.760274 −0.380137 0.924930i \(-0.624123\pi\)
−0.380137 + 0.924930i \(0.624123\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −14.0021 −1.71063 −0.855316 0.518106i \(-0.826637\pi\)
−0.855316 + 0.518106i \(0.826637\pi\)
\(68\) 0 0
\(69\) −6.36286 −0.765999
\(70\) 0 0
\(71\) −14.5305 −1.72445 −0.862225 0.506525i \(-0.830930\pi\)
−0.862225 + 0.506525i \(0.830930\pi\)
\(72\) 0 0
\(73\) 13.2216 1.54747 0.773733 0.633512i \(-0.218387\pi\)
0.773733 + 0.633512i \(0.218387\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.98466 0.335801 0.167900 0.985804i \(-0.446301\pi\)
0.167900 + 0.985804i \(0.446301\pi\)
\(80\) 0 0
\(81\) −11.1533 −1.23926
\(82\) 0 0
\(83\) −10.8923 −1.19558 −0.597791 0.801652i \(-0.703955\pi\)
−0.597791 + 0.801652i \(0.703955\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.23258 0.346569
\(88\) 0 0
\(89\) 4.71958 0.500274 0.250137 0.968210i \(-0.419524\pi\)
0.250137 + 0.968210i \(0.419524\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.56405 −0.162184
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.6575 1.08211 0.541053 0.840988i \(-0.318026\pi\)
0.541053 + 0.840988i \(0.318026\pi\)
\(98\) 0 0
\(99\) −4.23801 −0.425936
\(100\) 0 0
\(101\) −8.15552 −0.811505 −0.405753 0.913983i \(-0.632991\pi\)
−0.405753 + 0.913983i \(0.632991\pi\)
\(102\) 0 0
\(103\) −9.34463 −0.920753 −0.460377 0.887724i \(-0.652286\pi\)
−0.460377 + 0.887724i \(0.652286\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.81090 −0.561761 −0.280880 0.959743i \(-0.590626\pi\)
−0.280880 + 0.959743i \(0.590626\pi\)
\(108\) 0 0
\(109\) 14.7163 1.40957 0.704784 0.709422i \(-0.251044\pi\)
0.704784 + 0.709422i \(0.251044\pi\)
\(110\) 0 0
\(111\) −14.3293 −1.36008
\(112\) 0 0
\(113\) 0.0164044 0.00154319 0.000771596 1.00000i \(-0.499754\pi\)
0.000771596 1.00000i \(0.499754\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 3.45744 0.319640
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.70245 0.154769
\(122\) 0 0
\(123\) 14.4870 1.30625
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 15.9044 1.41128 0.705642 0.708569i \(-0.250659\pi\)
0.705642 + 0.708569i \(0.250659\pi\)
\(128\) 0 0
\(129\) −22.3892 −1.97126
\(130\) 0 0
\(131\) −18.2490 −1.59442 −0.797211 0.603701i \(-0.793692\pi\)
−0.797211 + 0.603701i \(0.793692\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −8.69992 −0.743284 −0.371642 0.928376i \(-0.621205\pi\)
−0.371642 + 0.928376i \(0.621205\pi\)
\(138\) 0 0
\(139\) −11.3728 −0.964625 −0.482313 0.875999i \(-0.660203\pi\)
−0.482313 + 0.875999i \(0.660203\pi\)
\(140\) 0 0
\(141\) −15.0967 −1.27137
\(142\) 0 0
\(143\) −10.3629 −0.866586
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −16.8009 −1.37639 −0.688194 0.725527i \(-0.741596\pi\)
−0.688194 + 0.725527i \(0.741596\pi\)
\(150\) 0 0
\(151\) 7.48593 0.609196 0.304598 0.952481i \(-0.401478\pi\)
0.304598 + 0.952481i \(0.401478\pi\)
\(152\) 0 0
\(153\) 4.42168 0.357472
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.7032 1.17344 0.586720 0.809790i \(-0.300419\pi\)
0.586720 + 0.809790i \(0.300419\pi\)
\(158\) 0 0
\(159\) 6.81197 0.540224
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 6.39354 0.500781 0.250390 0.968145i \(-0.419441\pi\)
0.250390 + 0.968145i \(0.419441\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 15.8509 1.22658 0.613291 0.789857i \(-0.289845\pi\)
0.613291 + 0.789857i \(0.289845\pi\)
\(168\) 0 0
\(169\) −4.54581 −0.349678
\(170\) 0 0
\(171\) −4.25625 −0.325484
\(172\) 0 0
\(173\) −7.25082 −0.551269 −0.275635 0.961262i \(-0.588888\pi\)
−0.275635 + 0.961262i \(0.588888\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 16.6608 1.25230
\(178\) 0 0
\(179\) 13.5998 1.01649 0.508247 0.861211i \(-0.330294\pi\)
0.508247 + 0.861211i \(0.330294\pi\)
\(180\) 0 0
\(181\) 16.5469 1.22992 0.614960 0.788558i \(-0.289172\pi\)
0.614960 + 0.788558i \(0.289172\pi\)
\(182\) 0 0
\(183\) −12.1533 −0.898401
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −13.2530 −0.969152
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.63043 −0.624476 −0.312238 0.950004i \(-0.601079\pi\)
−0.312238 + 0.950004i \(0.601079\pi\)
\(192\) 0 0
\(193\) −13.6739 −0.984270 −0.492135 0.870519i \(-0.663783\pi\)
−0.492135 + 0.870519i \(0.663783\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −26.8922 −1.91599 −0.957996 0.286783i \(-0.907414\pi\)
−0.957996 + 0.286783i \(0.907414\pi\)
\(198\) 0 0
\(199\) 3.68493 0.261218 0.130609 0.991434i \(-0.458307\pi\)
0.130609 + 0.991434i \(0.458307\pi\)
\(200\) 0 0
\(201\) −28.6586 −2.02142
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −3.69667 −0.256936
\(208\) 0 0
\(209\) 12.7571 0.882428
\(210\) 0 0
\(211\) 4.91193 0.338151 0.169075 0.985603i \(-0.445922\pi\)
0.169075 + 0.985603i \(0.445922\pi\)
\(212\) 0 0
\(213\) −29.7399 −2.03775
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 27.0609 1.82861
\(220\) 0 0
\(221\) 10.8120 0.727292
\(222\) 0 0
\(223\) 12.6575 0.847609 0.423805 0.905754i \(-0.360694\pi\)
0.423805 + 0.905754i \(0.360694\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.94007 0.527001 0.263500 0.964659i \(-0.415123\pi\)
0.263500 + 0.964659i \(0.415123\pi\)
\(228\) 0 0
\(229\) −6.64039 −0.438809 −0.219405 0.975634i \(-0.570411\pi\)
−0.219405 + 0.975634i \(0.570411\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.7042 0.897794 0.448897 0.893583i \(-0.351817\pi\)
0.448897 + 0.893583i \(0.351817\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 6.10880 0.396809
\(238\) 0 0
\(239\) 28.0010 1.81124 0.905618 0.424095i \(-0.139408\pi\)
0.905618 + 0.424095i \(0.139408\pi\)
\(240\) 0 0
\(241\) −22.8477 −1.47175 −0.735874 0.677118i \(-0.763228\pi\)
−0.735874 + 0.677118i \(0.763228\pi\)
\(242\) 0 0
\(243\) −11.7086 −0.751107
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −10.4075 −0.662211
\(248\) 0 0
\(249\) −22.2935 −1.41279
\(250\) 0 0
\(251\) −3.55973 −0.224688 −0.112344 0.993669i \(-0.535836\pi\)
−0.112344 + 0.993669i \(0.535836\pi\)
\(252\) 0 0
\(253\) 11.0799 0.696588
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 27.6239 1.72313 0.861567 0.507644i \(-0.169484\pi\)
0.861567 + 0.507644i \(0.169484\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1.87805 0.116248
\(262\) 0 0
\(263\) −25.2226 −1.55529 −0.777647 0.628701i \(-0.783587\pi\)
−0.777647 + 0.628701i \(0.783587\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 9.65969 0.591164
\(268\) 0 0
\(269\) −16.6389 −1.01449 −0.507246 0.861801i \(-0.669337\pi\)
−0.507246 + 0.861801i \(0.669337\pi\)
\(270\) 0 0
\(271\) −24.5262 −1.48986 −0.744929 0.667144i \(-0.767517\pi\)
−0.744929 + 0.667144i \(0.767517\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −31.5793 −1.89742 −0.948709 0.316150i \(-0.897610\pi\)
−0.948709 + 0.316150i \(0.897610\pi\)
\(278\) 0 0
\(279\) −0.908675 −0.0544010
\(280\) 0 0
\(281\) −28.0413 −1.67280 −0.836402 0.548117i \(-0.815345\pi\)
−0.836402 + 0.548117i \(0.815345\pi\)
\(282\) 0 0
\(283\) −29.6707 −1.76374 −0.881869 0.471495i \(-0.843715\pi\)
−0.881869 + 0.471495i \(0.843715\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −3.17270 −0.186629
\(290\) 0 0
\(291\) 21.8130 1.27870
\(292\) 0 0
\(293\) −9.43031 −0.550925 −0.275462 0.961312i \(-0.588831\pi\)
−0.275462 + 0.961312i \(0.588831\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 13.2099 0.766515
\(298\) 0 0
\(299\) −9.03916 −0.522748
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −16.6922 −0.958939
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −4.59798 −0.262420 −0.131210 0.991355i \(-0.541886\pi\)
−0.131210 + 0.991355i \(0.541886\pi\)
\(308\) 0 0
\(309\) −19.1259 −1.08804
\(310\) 0 0
\(311\) −1.48055 −0.0839540 −0.0419770 0.999119i \(-0.513366\pi\)
−0.0419770 + 0.999119i \(0.513366\pi\)
\(312\) 0 0
\(313\) −17.1402 −0.968821 −0.484410 0.874841i \(-0.660966\pi\)
−0.484410 + 0.874841i \(0.660966\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.86301 −0.385465 −0.192733 0.981251i \(-0.561735\pi\)
−0.192733 + 0.981251i \(0.561735\pi\)
\(318\) 0 0
\(319\) −5.62902 −0.315164
\(320\) 0 0
\(321\) −11.8933 −0.663821
\(322\) 0 0
\(323\) −13.3100 −0.740587
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 30.1203 1.66566
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −3.41736 −0.187835 −0.0939176 0.995580i \(-0.529939\pi\)
−0.0939176 + 0.995580i \(0.529939\pi\)
\(332\) 0 0
\(333\) −8.32497 −0.456206
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 26.3574 1.43578 0.717890 0.696156i \(-0.245108\pi\)
0.717890 + 0.696156i \(0.245108\pi\)
\(338\) 0 0
\(339\) 0.0335753 0.00182356
\(340\) 0 0
\(341\) 2.72354 0.147488
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.54653 0.512485 0.256242 0.966613i \(-0.417516\pi\)
0.256242 + 0.966613i \(0.417516\pi\)
\(348\) 0 0
\(349\) −10.8259 −0.579496 −0.289748 0.957103i \(-0.593572\pi\)
−0.289748 + 0.957103i \(0.593572\pi\)
\(350\) 0 0
\(351\) −10.7768 −0.575224
\(352\) 0 0
\(353\) 20.6443 1.09879 0.549393 0.835564i \(-0.314859\pi\)
0.549393 + 0.835564i \(0.314859\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 21.6860 1.14454 0.572272 0.820064i \(-0.306062\pi\)
0.572272 + 0.820064i \(0.306062\pi\)
\(360\) 0 0
\(361\) −6.18798 −0.325683
\(362\) 0 0
\(363\) 3.48446 0.182887
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −18.8120 −0.981977 −0.490988 0.871166i \(-0.663364\pi\)
−0.490988 + 0.871166i \(0.663364\pi\)
\(368\) 0 0
\(369\) 8.41660 0.438150
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 28.1818 1.45920 0.729600 0.683874i \(-0.239706\pi\)
0.729600 + 0.683874i \(0.239706\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.59224 0.236512
\(378\) 0 0
\(379\) −6.93397 −0.356174 −0.178087 0.984015i \(-0.556991\pi\)
−0.178087 + 0.984015i \(0.556991\pi\)
\(380\) 0 0
\(381\) 32.5519 1.66769
\(382\) 0 0
\(383\) 15.1488 0.774069 0.387034 0.922065i \(-0.373499\pi\)
0.387034 + 0.922065i \(0.373499\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −13.0076 −0.661212
\(388\) 0 0
\(389\) 26.1606 1.32639 0.663196 0.748445i \(-0.269199\pi\)
0.663196 + 0.748445i \(0.269199\pi\)
\(390\) 0 0
\(391\) −11.5601 −0.584619
\(392\) 0 0
\(393\) −37.3507 −1.88409
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −20.0561 −1.00659 −0.503293 0.864116i \(-0.667878\pi\)
−0.503293 + 0.864116i \(0.667878\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 11.4509 0.571833 0.285916 0.958255i \(-0.407702\pi\)
0.285916 + 0.958255i \(0.407702\pi\)
\(402\) 0 0
\(403\) −2.22191 −0.110681
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.9522 1.23683
\(408\) 0 0
\(409\) −26.6218 −1.31636 −0.658182 0.752859i \(-0.728674\pi\)
−0.658182 + 0.752859i \(0.728674\pi\)
\(410\) 0 0
\(411\) −17.8064 −0.878324
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −23.2770 −1.13988
\(418\) 0 0
\(419\) 26.8944 1.31388 0.656939 0.753944i \(-0.271851\pi\)
0.656939 + 0.753944i \(0.271851\pi\)
\(420\) 0 0
\(421\) −19.2662 −0.938975 −0.469487 0.882939i \(-0.655561\pi\)
−0.469487 + 0.882939i \(0.655561\pi\)
\(422\) 0 0
\(423\) −8.77082 −0.426452
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −21.2100 −1.02403
\(430\) 0 0
\(431\) 20.2355 0.974709 0.487354 0.873204i \(-0.337962\pi\)
0.487354 + 0.873204i \(0.337962\pi\)
\(432\) 0 0
\(433\) 3.13135 0.150483 0.0752416 0.997165i \(-0.476027\pi\)
0.0752416 + 0.997165i \(0.476027\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 11.1276 0.532305
\(438\) 0 0
\(439\) −17.4585 −0.833247 −0.416623 0.909079i \(-0.636787\pi\)
−0.416623 + 0.909079i \(0.636787\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 13.5619 0.644344 0.322172 0.946681i \(-0.395587\pi\)
0.322172 + 0.946681i \(0.395587\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −34.3870 −1.62645
\(448\) 0 0
\(449\) −28.7260 −1.35566 −0.677832 0.735216i \(-0.737081\pi\)
−0.677832 + 0.735216i \(0.737081\pi\)
\(450\) 0 0
\(451\) −25.2268 −1.18788
\(452\) 0 0
\(453\) 15.3217 0.719875
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8.23873 −0.385392 −0.192696 0.981259i \(-0.561723\pi\)
−0.192696 + 0.981259i \(0.561723\pi\)
\(458\) 0 0
\(459\) −13.7824 −0.643305
\(460\) 0 0
\(461\) −29.8729 −1.39132 −0.695660 0.718371i \(-0.744888\pi\)
−0.695660 + 0.718371i \(0.744888\pi\)
\(462\) 0 0
\(463\) 11.4155 0.530525 0.265262 0.964176i \(-0.414541\pi\)
0.265262 + 0.964176i \(0.414541\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 25.2223 1.16715 0.583574 0.812060i \(-0.301654\pi\)
0.583574 + 0.812060i \(0.301654\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 30.0934 1.38663
\(472\) 0 0
\(473\) 38.9871 1.79263
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3.95759 0.181206
\(478\) 0 0
\(479\) 27.0788 1.23726 0.618632 0.785681i \(-0.287687\pi\)
0.618632 + 0.785681i \(0.287687\pi\)
\(480\) 0 0
\(481\) −20.3564 −0.928170
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 18.3464 0.831355 0.415678 0.909512i \(-0.363544\pi\)
0.415678 + 0.909512i \(0.363544\pi\)
\(488\) 0 0
\(489\) 13.0858 0.591762
\(490\) 0 0
\(491\) −25.4457 −1.14835 −0.574173 0.818734i \(-0.694676\pi\)
−0.574173 + 0.818734i \(0.694676\pi\)
\(492\) 0 0
\(493\) 5.87297 0.264505
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 4.14309 0.185470 0.0927350 0.995691i \(-0.470439\pi\)
0.0927350 + 0.995691i \(0.470439\pi\)
\(500\) 0 0
\(501\) 32.4426 1.44943
\(502\) 0 0
\(503\) −19.0099 −0.847610 −0.423805 0.905754i \(-0.639306\pi\)
−0.423805 + 0.905754i \(0.639306\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −9.30405 −0.413207
\(508\) 0 0
\(509\) −3.84589 −0.170466 −0.0852331 0.996361i \(-0.527163\pi\)
−0.0852331 + 0.996361i \(0.527163\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 13.2667 0.585740
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 26.2885 1.15617
\(518\) 0 0
\(519\) −14.8405 −0.651424
\(520\) 0 0
\(521\) 13.6154 0.596504 0.298252 0.954487i \(-0.403596\pi\)
0.298252 + 0.954487i \(0.403596\pi\)
\(522\) 0 0
\(523\) −22.1673 −0.969307 −0.484653 0.874706i \(-0.661054\pi\)
−0.484653 + 0.874706i \(0.661054\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.84157 −0.123781
\(528\) 0 0
\(529\) −13.3354 −0.579800
\(530\) 0 0
\(531\) 9.67950 0.420054
\(532\) 0 0
\(533\) 20.5804 0.891436
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 27.8350 1.20117
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 29.0149 1.24745 0.623725 0.781644i \(-0.285619\pi\)
0.623725 + 0.781644i \(0.285619\pi\)
\(542\) 0 0
\(543\) 33.8670 1.45337
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 28.6053 1.22308 0.611538 0.791215i \(-0.290551\pi\)
0.611538 + 0.791215i \(0.290551\pi\)
\(548\) 0 0
\(549\) −7.06080 −0.301348
\(550\) 0 0
\(551\) −5.65324 −0.240836
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 31.6304 1.34022 0.670111 0.742261i \(-0.266246\pi\)
0.670111 + 0.742261i \(0.266246\pi\)
\(558\) 0 0
\(559\) −31.8063 −1.34526
\(560\) 0 0
\(561\) −27.1252 −1.14523
\(562\) 0 0
\(563\) −20.0110 −0.843364 −0.421682 0.906744i \(-0.638560\pi\)
−0.421682 + 0.906744i \(0.638560\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 35.3446 1.48172 0.740861 0.671659i \(-0.234418\pi\)
0.740861 + 0.671659i \(0.234418\pi\)
\(570\) 0 0
\(571\) −15.6707 −0.655797 −0.327898 0.944713i \(-0.606340\pi\)
−0.327898 + 0.944713i \(0.606340\pi\)
\(572\) 0 0
\(573\) −17.6642 −0.737931
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −25.9057 −1.07847 −0.539235 0.842156i \(-0.681286\pi\)
−0.539235 + 0.842156i \(0.681286\pi\)
\(578\) 0 0
\(579\) −27.9868 −1.16309
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −11.8619 −0.491272
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −38.2409 −1.57837 −0.789185 0.614156i \(-0.789497\pi\)
−0.789185 + 0.614156i \(0.789497\pi\)
\(588\) 0 0
\(589\) 2.73526 0.112705
\(590\) 0 0
\(591\) −55.0411 −2.26409
\(592\) 0 0
\(593\) −46.2697 −1.90007 −0.950034 0.312148i \(-0.898952\pi\)
−0.950034 + 0.312148i \(0.898952\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 7.54206 0.308676
\(598\) 0 0
\(599\) −21.0525 −0.860180 −0.430090 0.902786i \(-0.641518\pi\)
−0.430090 + 0.902786i \(0.641518\pi\)
\(600\) 0 0
\(601\) 27.5283 1.12290 0.561451 0.827510i \(-0.310243\pi\)
0.561451 + 0.827510i \(0.310243\pi\)
\(602\) 0 0
\(603\) −16.6499 −0.678038
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 21.6514 0.878802 0.439401 0.898291i \(-0.355191\pi\)
0.439401 + 0.898291i \(0.355191\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −21.4466 −0.867635
\(612\) 0 0
\(613\) 26.2336 1.05956 0.529782 0.848134i \(-0.322274\pi\)
0.529782 + 0.848134i \(0.322274\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9.48553 0.381873 0.190937 0.981602i \(-0.438848\pi\)
0.190937 + 0.981602i \(0.438848\pi\)
\(618\) 0 0
\(619\) 6.13942 0.246764 0.123382 0.992359i \(-0.460626\pi\)
0.123382 + 0.992359i \(0.460626\pi\)
\(620\) 0 0
\(621\) 11.5225 0.462382
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 26.1104 1.04275
\(628\) 0 0
\(629\) −26.0335 −1.03802
\(630\) 0 0
\(631\) 36.8070 1.46527 0.732633 0.680624i \(-0.238291\pi\)
0.732633 + 0.680624i \(0.238291\pi\)
\(632\) 0 0
\(633\) 10.0534 0.399586
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −17.2782 −0.683515
\(640\) 0 0
\(641\) −19.6440 −0.775892 −0.387946 0.921682i \(-0.626815\pi\)
−0.387946 + 0.921682i \(0.626815\pi\)
\(642\) 0 0
\(643\) 17.8590 0.704292 0.352146 0.935945i \(-0.385452\pi\)
0.352146 + 0.935945i \(0.385452\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 23.4303 0.921141 0.460570 0.887623i \(-0.347645\pi\)
0.460570 + 0.887623i \(0.347645\pi\)
\(648\) 0 0
\(649\) −29.0120 −1.13882
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.04348 0.119101 0.0595503 0.998225i \(-0.481033\pi\)
0.0595503 + 0.998225i \(0.481033\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 15.7218 0.613364
\(658\) 0 0
\(659\) −0.576904 −0.0224730 −0.0112365 0.999937i \(-0.503577\pi\)
−0.0112365 + 0.999937i \(0.503577\pi\)
\(660\) 0 0
\(661\) −28.1584 −1.09523 −0.547617 0.836729i \(-0.684465\pi\)
−0.547617 + 0.836729i \(0.684465\pi\)
\(662\) 0 0
\(663\) 22.1292 0.859426
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −4.90999 −0.190116
\(668\) 0 0
\(669\) 25.9065 1.00160
\(670\) 0 0
\(671\) 21.1631 0.816992
\(672\) 0 0
\(673\) −17.7835 −0.685503 −0.342751 0.939426i \(-0.611359\pi\)
−0.342751 + 0.939426i \(0.611359\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 32.6768 1.25587 0.627936 0.778265i \(-0.283900\pi\)
0.627936 + 0.778265i \(0.283900\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 16.2512 0.622746
\(682\) 0 0
\(683\) 16.1351 0.617393 0.308696 0.951161i \(-0.400107\pi\)
0.308696 + 0.951161i \(0.400107\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −13.5911 −0.518532
\(688\) 0 0
\(689\) 9.67716 0.368671
\(690\) 0 0
\(691\) 16.1862 0.615750 0.307875 0.951427i \(-0.400382\pi\)
0.307875 + 0.951427i \(0.400382\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 26.3201 0.996943
\(698\) 0 0
\(699\) 28.0489 1.06091
\(700\) 0 0
\(701\) 19.8684 0.750419 0.375210 0.926940i \(-0.377571\pi\)
0.375210 + 0.926940i \(0.377571\pi\)
\(702\) 0 0
\(703\) 25.0595 0.945138
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −24.7667 −0.930131 −0.465066 0.885276i \(-0.653969\pi\)
−0.465066 + 0.885276i \(0.653969\pi\)
\(710\) 0 0
\(711\) 3.54906 0.133100
\(712\) 0 0
\(713\) 2.37565 0.0889688
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 57.3105 2.14030
\(718\) 0 0
\(719\) −7.44692 −0.277723 −0.138862 0.990312i \(-0.544344\pi\)
−0.138862 + 0.990312i \(0.544344\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −46.7630 −1.73913
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −2.63572 −0.0977534 −0.0488767 0.998805i \(-0.515564\pi\)
−0.0488767 + 0.998805i \(0.515564\pi\)
\(728\) 0 0
\(729\) 9.49568 0.351692
\(730\) 0 0
\(731\) −40.6768 −1.50448
\(732\) 0 0
\(733\) 12.3811 0.457305 0.228652 0.973508i \(-0.426568\pi\)
0.228652 + 0.973508i \(0.426568\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 49.9043 1.83825
\(738\) 0 0
\(739\) 6.73025 0.247576 0.123788 0.992309i \(-0.460496\pi\)
0.123788 + 0.992309i \(0.460496\pi\)
\(740\) 0 0
\(741\) −21.3012 −0.782521
\(742\) 0 0
\(743\) 18.6268 0.683352 0.341676 0.939818i \(-0.389005\pi\)
0.341676 + 0.939818i \(0.389005\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −12.9520 −0.473889
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −41.4245 −1.51160 −0.755801 0.654801i \(-0.772752\pi\)
−0.755801 + 0.654801i \(0.772752\pi\)
\(752\) 0 0
\(753\) −7.28581 −0.265510
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 27.3296 0.993311 0.496655 0.867948i \(-0.334561\pi\)
0.496655 + 0.867948i \(0.334561\pi\)
\(758\) 0 0
\(759\) 22.6776 0.823143
\(760\) 0 0
\(761\) 13.5651 0.491735 0.245868 0.969303i \(-0.420927\pi\)
0.245868 + 0.969303i \(0.420927\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 23.6685 0.854619
\(768\) 0 0
\(769\) −33.6618 −1.21388 −0.606938 0.794749i \(-0.707602\pi\)
−0.606938 + 0.794749i \(0.707602\pi\)
\(770\) 0 0
\(771\) 56.5387 2.03619
\(772\) 0 0
\(773\) 1.06100 0.0381615 0.0190808 0.999818i \(-0.493926\pi\)
0.0190808 + 0.999818i \(0.493926\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −25.3353 −0.907733
\(780\) 0 0
\(781\) 51.7873 1.85310
\(782\) 0 0
\(783\) −5.85388 −0.209200
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −40.3760 −1.43925 −0.719625 0.694363i \(-0.755686\pi\)
−0.719625 + 0.694363i \(0.755686\pi\)
\(788\) 0 0
\(789\) −51.6239 −1.83786
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −17.2652 −0.613105
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −6.16436 −0.218353 −0.109176 0.994022i \(-0.534821\pi\)
−0.109176 + 0.994022i \(0.534821\pi\)
\(798\) 0 0
\(799\) −27.4278 −0.970325
\(800\) 0 0
\(801\) 5.61205 0.198292
\(802\) 0 0
\(803\) −47.1223 −1.66291
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −34.0554 −1.19881
\(808\) 0 0
\(809\) −28.0767 −0.987123 −0.493561 0.869711i \(-0.664305\pi\)
−0.493561 + 0.869711i \(0.664305\pi\)
\(810\) 0 0
\(811\) −40.3175 −1.41574 −0.707869 0.706343i \(-0.750344\pi\)
−0.707869 + 0.706343i \(0.750344\pi\)
\(812\) 0 0
\(813\) −50.1984 −1.76053
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 39.1549 1.36986
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 54.6600 1.90765 0.953823 0.300368i \(-0.0971094\pi\)
0.953823 + 0.300368i \(0.0971094\pi\)
\(822\) 0 0
\(823\) 25.0682 0.873821 0.436911 0.899505i \(-0.356073\pi\)
0.436911 + 0.899505i \(0.356073\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 19.8101 0.688866 0.344433 0.938811i \(-0.388071\pi\)
0.344433 + 0.938811i \(0.388071\pi\)
\(828\) 0 0
\(829\) 20.2693 0.703982 0.351991 0.936003i \(-0.385505\pi\)
0.351991 + 0.936003i \(0.385505\pi\)
\(830\) 0 0
\(831\) −64.6344 −2.24214
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 2.83234 0.0978999
\(838\) 0 0
\(839\) 54.3873 1.87766 0.938829 0.344384i \(-0.111912\pi\)
0.938829 + 0.344384i \(0.111912\pi\)
\(840\) 0 0
\(841\) −26.5055 −0.913984
\(842\) 0 0
\(843\) −57.3929 −1.97672
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −60.7278 −2.08417
\(850\) 0 0
\(851\) 21.7649 0.746091
\(852\) 0 0
\(853\) −3.88160 −0.132903 −0.0664517 0.997790i \(-0.521168\pi\)
−0.0664517 + 0.997790i \(0.521168\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.1826 0.962699 0.481350 0.876529i \(-0.340147\pi\)
0.481350 + 0.876529i \(0.340147\pi\)
\(858\) 0 0
\(859\) 18.5009 0.631241 0.315621 0.948885i \(-0.397787\pi\)
0.315621 + 0.948885i \(0.397787\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 14.0653 0.478787 0.239394 0.970923i \(-0.423051\pi\)
0.239394 + 0.970923i \(0.423051\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −6.49365 −0.220536
\(868\) 0 0
\(869\) −10.6375 −0.360852
\(870\) 0 0
\(871\) −40.7127 −1.37950
\(872\) 0 0
\(873\) 12.6728 0.428911
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 13.5677 0.458147 0.229074 0.973409i \(-0.426430\pi\)
0.229074 + 0.973409i \(0.426430\pi\)
\(878\) 0 0
\(879\) −19.3013 −0.651017
\(880\) 0 0
\(881\) 9.90324 0.333649 0.166824 0.985987i \(-0.446649\pi\)
0.166824 + 0.985987i \(0.446649\pi\)
\(882\) 0 0
\(883\) 27.4705 0.924456 0.462228 0.886761i \(-0.347050\pi\)
0.462228 + 0.886761i \(0.347050\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −35.8458 −1.20359 −0.601793 0.798652i \(-0.705547\pi\)
−0.601793 + 0.798652i \(0.705547\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 39.7511 1.33171
\(892\) 0 0
\(893\) 26.4016 0.883497
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −18.5007 −0.617721
\(898\) 0 0
\(899\) −1.20692 −0.0402531
\(900\) 0 0
\(901\) 12.3760 0.412305
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 5.23695 0.173890 0.0869450 0.996213i \(-0.472290\pi\)
0.0869450 + 0.996213i \(0.472290\pi\)
\(908\) 0 0
\(909\) −9.69774 −0.321654
\(910\) 0 0
\(911\) −13.2519 −0.439054 −0.219527 0.975606i \(-0.570451\pi\)
−0.219527 + 0.975606i \(0.570451\pi\)
\(912\) 0 0
\(913\) 38.8206 1.28477
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 44.6399 1.47254 0.736268 0.676690i \(-0.236586\pi\)
0.736268 + 0.676690i \(0.236586\pi\)
\(920\) 0 0
\(921\) −9.41081 −0.310097
\(922\) 0 0
\(923\) −42.2489 −1.39064
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −11.1117 −0.364956
\(928\) 0 0
\(929\) 8.36215 0.274353 0.137177 0.990547i \(-0.456197\pi\)
0.137177 + 0.990547i \(0.456197\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −3.03027 −0.0992067
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −2.76234 −0.0902416 −0.0451208 0.998982i \(-0.514367\pi\)
−0.0451208 + 0.998982i \(0.514367\pi\)
\(938\) 0 0
\(939\) −35.0813 −1.14484
\(940\) 0 0
\(941\) −15.7224 −0.512536 −0.256268 0.966606i \(-0.582493\pi\)
−0.256268 + 0.966606i \(0.582493\pi\)
\(942\) 0 0
\(943\) −22.0044 −0.716563
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 11.6932 0.379979 0.189989 0.981786i \(-0.439155\pi\)
0.189989 + 0.981786i \(0.439155\pi\)
\(948\) 0 0
\(949\) 38.4431 1.24792
\(950\) 0 0
\(951\) −14.0467 −0.455496
\(952\) 0 0
\(953\) −3.83507 −0.124230 −0.0621151 0.998069i \(-0.519785\pi\)
−0.0621151 + 0.998069i \(0.519785\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −11.5211 −0.372423
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −30.4160 −0.981163
\(962\) 0 0
\(963\) −6.90974 −0.222663
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 37.3745 1.20188 0.600942 0.799292i \(-0.294792\pi\)
0.600942 + 0.799292i \(0.294792\pi\)
\(968\) 0 0
\(969\) −27.2419 −0.875137
\(970\) 0 0
\(971\) 44.5783 1.43058 0.715292 0.698825i \(-0.246294\pi\)
0.715292 + 0.698825i \(0.246294\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −12.7664 −0.408432 −0.204216 0.978926i \(-0.565464\pi\)
−0.204216 + 0.978926i \(0.565464\pi\)
\(978\) 0 0
\(979\) −16.8208 −0.537595
\(980\) 0 0
\(981\) 17.4992 0.558706
\(982\) 0 0
\(983\) 15.7641 0.502797 0.251399 0.967884i \(-0.419109\pi\)
0.251399 + 0.967884i \(0.419109\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 34.0071 1.08136
\(990\) 0 0
\(991\) −1.57162 −0.0499241 −0.0249620 0.999688i \(-0.507946\pi\)
−0.0249620 + 0.999688i \(0.507946\pi\)
\(992\) 0 0
\(993\) −6.99441 −0.221961
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 22.7367 0.720078 0.360039 0.932937i \(-0.382763\pi\)
0.360039 + 0.932937i \(0.382763\pi\)
\(998\) 0 0
\(999\) 25.9489 0.820987
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9800.2.a.cj.1.4 4
5.4 even 2 9800.2.a.ct.1.1 4
7.2 even 3 1400.2.q.m.1201.1 yes 8
7.4 even 3 1400.2.q.m.401.1 yes 8
7.6 odd 2 9800.2.a.cu.1.1 4
35.2 odd 12 1400.2.bh.j.249.7 16
35.4 even 6 1400.2.q.l.401.4 8
35.9 even 6 1400.2.q.l.1201.4 yes 8
35.18 odd 12 1400.2.bh.j.849.7 16
35.23 odd 12 1400.2.bh.j.249.2 16
35.32 odd 12 1400.2.bh.j.849.2 16
35.34 odd 2 9800.2.a.ck.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1400.2.q.l.401.4 8 35.4 even 6
1400.2.q.l.1201.4 yes 8 35.9 even 6
1400.2.q.m.401.1 yes 8 7.4 even 3
1400.2.q.m.1201.1 yes 8 7.2 even 3
1400.2.bh.j.249.2 16 35.23 odd 12
1400.2.bh.j.249.7 16 35.2 odd 12
1400.2.bh.j.849.2 16 35.32 odd 12
1400.2.bh.j.849.7 16 35.18 odd 12
9800.2.a.cj.1.4 4 1.1 even 1 trivial
9800.2.a.ck.1.4 4 35.34 odd 2
9800.2.a.ct.1.1 4 5.4 even 2
9800.2.a.cu.1.1 4 7.6 odd 2